which is, up to constants, the result (12) of [1]} when one takes \(\Gamma _{m}\)  in [1]} as the collection of piecewise constant functions on \({{W}}\) . The difference is their model selection procedure, on the one hand, does not require Assumption REF  to be satisfied and can be applied to other types of models to approximate the potential \({\gamma }^{*}\)  besides piecewise constant ones. On the other hand, when the number of models becomes large, model selection strategy is more of theoretical interest due to its expensive numerical cost. Our estimator selection strategy, however, allows to deal with random partitions which can be obtained for example from dynamic programming algorithm (e.g. [3]}) or CART algorithm (e.g. [4]}). Efficiently reducing the cardinality of \(\widehat{\Gamma }\) , these algorithms together with our estimator selection procedure take the model selection strategy into practice. Moreover, the idea that selecting among random candidates set makes the selection between estimators given by different model selection strategies possible.
